Title & Abstract

Title: A new numerical method for parameter sensitivity analysis with application to Greeks computatation in a general stochastic volatility model

Abstract: The probabilistic representation of parameter differentiations of diffusion semigroups using Malliavin calculus provides a kind of automatic differentiation method and is useful to estimate Greeks such as delta and vega in finance. In the talk, we introduce a second order discretization method for parameter differentiations in a general stochastic volatility model. A new algorithm is provided with an efficient simulation method. The effectiveness of the proposed method is checked through numerical examples under lognormal SABR model and Heston model. The talk is based on a joint work with Toshihiro Yamada.

Title: Trading constraints in continuous-time Kyle models

Abstract: In a continuous-time Kyle setting, we prove global existence of an equilibrium when the insider faces a terminal trading constraint. We prove that our equilibrium model produces output consistent with several empirical stylized facts such as autocorrelated aggregate holdings, decreasing price impacts over the trading day, and U shaped optimal trading patterns. This work is based on a collaboration with Heeyoung Kwon and Kasper Larsen.

Title: Optimal maximization with random time in an illiquid market with transaction costs and search frictions

Abstract: In this paper, we consider an optimal investment problem to maximize expected power-utility of the random terminal wealth in an market with two types of illiquidity: transaction costs and search frictions. In the market model, we suppose that an investor can trade only at arrival times of a Poisson process, and pays proportional transaction costs for purchasing or selling stocks. Furthermore, the random terminal time is exponentially distributed which is independent of the Poisson process and Brownian motion. We characterize a unique optimal trading strategy in terms of buy region, no-trade region, and sell region. Furthermore, we provide asymptotic expansions on small transaction costs and small search frictions for boundaries of the no-trade region and value function. The asymptotic expansions on small transaction costs and small search frictions show that the width of no-trade region widens and the diminishing effect of the value function increases as the transaction costs increase and the search frictions decrease. Moreover, the first order of the width of no-trade region and value reduction are represented by transaction costs parameter times search frictions parameter and transaction costs parameter times square root of search frictions parameter, respectively. This work is based on a collaboration with Jin Hyuk Choi.

Title: Back to The Future: A Deep Propagator Approach to Estimate The Market Expectation

Abstract: In this study, we consider a contemporary buy-side problem, which is estimating the market forecast of return and its associated natural probability measure from the observation of option prices. Though Ross shed some light on the issue by proposing the so-called recovery theorem, his model is considered highly restrictive with empirical evidences. Therefore, we develop a more natural and flexible framework such that we do not need to enforce a heavy set of assumptions, including the transition independence of the pricing kernel as in Ross-type models. Instead, we only assume the propagation property of the natural density and illustrate its functional form through a neural network. We call such model the “deep propagator” and show how to recover the market expectation from the option prices with our non-parametric approach. Utilizing 10 years of options data on the S&P500, the numerical analysis shows some promising results and implies the great potential of using the deep propagator model to estimate the market expectation. This work is based on a collaboration with Jiro Akahori.

Title: A Discrete Scheme of Static Hedging of Barrier Options

Abstract: We consider a discrete scheme for static hedging of barrier options, by establishing a discrete version of the transform which Peter Carr and Sergey Nadtochiy (2013) introduced, for a general one dimensional diffusion case. The transform describes the (put type) pay-off which balances at the barrier with a given (call-type) pay-off and hence the plain option with the former pay-off statically hedges a barrier option with the latter pay-off. In this talk we will construct the map for a class of Markov chains, which includes a discretization of Carr-Nadtochiy’s correspondence, and also its multi-dimensional version.

Title: An equilibrium model of production and capacity expansion

Abstract: We consider a model with producers making decisions on how much to produce and how much to invest in expansion of capacity (or capability) of future production. We initially focus on the single agent's problem, considered to be a price-taker, and given exogenous prices. A characterisation of the solution in terms of a backwards SDE is given. We then consider demand functions exogenously given, and a multi-agent problem where prices are formed within equilibrium. The latter is characterised via the solution of a coupled forward-backward SDE system. We reduce the previous system to a second-order non-linear ODE. (Based on joint work-in-progress with Junchao Jia and Michael Zervos.)

Title: Mind the gap in the mining game

Abstract: In a blockchain system that uses a proof-of-work based consensus protocol, such as Bitcoin, a miner’s revenue consists of block rewards and transaction fees. Bitcoin’s block reward is halved over time, ultimately converging to zero. Previous studies have shown that if the block reward is small enough, then there might be a mining gap in which miners opt not to operate in certain periods of time. In this paper, our primary focus is on miners’ optimal operating strategies in a competitive setting, characterization of the conditions for mining gap, and their consequences.

Title: Optimal Consumption and Investment with Welfare Constraints

Abstract: In this study, we investigate an optimal consumption and investment problem of an economic agent who faces a welfare constraint; the agent does not accept her expected utility (continuation value) falls below a certain fixed level regardless of the time and state. This optimization problem involves an infinite number of constraints. Using a duality approach, we transform infinitely many constraints into a single constraint and define the dual problem, which becomes a two-dimensional singular control problem. The dual problem provides its associated Hamilton-Jacobi-Bellman (HJB) equation with a gradient constraint. Under a general class of utility functions, we obtain an explicit solution to the HJB equation and provide optimal strategies by establishing a duality theorem. As an example, we consider hyperbolic absolute risk aversion (HARA) utility, which may incorporate a government subsidy or a basic support, and provide the solution and its implications. This work is based on a collaboration with Junkee Jeon.

Title: Equity Investing in Inflationary Macro Regime

Abstract: The inflationary environment that has hit for the first time in 40 years is providing a very unfamiliar investment environment for the current generation of investors. In the midst of such changes in the macro environment, many research papers on investment strategies and risk premium have been published recently. We will review recent research trends on the subject and explain the situation in 2022, when inflation intensifies in earnest, by region in terms of equity factor investing.

Title: Recursive Preference and Time-varying Borrowing Constraints

Abstract: We study a continuous-time optimal consumption and portfolio selection problem when an economic agent with recursive utility has stochastic income and time-varying borrowing constraints. The optimal portfolio depends on the elasticity of intertemporal substitution (EIS) due to the borrowing constraints even if the investment opportunity is constant. The paper has novel implications for the optimal policies and the marginal propensity to consume (MPC) under recursive preference. The model provides a testable implication that stock market participants have higher MPCs than non-participants. We also make a technical contribution by developing a new transform to handle the problems with recursive utility. This work is based on a collaboration with Kyoung Jin Choi and Minsuk Kwak.

Title: A representative agent model based on risk-neutral prices

Abstract: In this study, we present a method for modeling the utility function of a representative agent in the consumption-based capital asset pricing model (CAPM). We derive an analytic model for the utility function from the risk-neutral information of a state variable. Our methodology is based on the concept used in the Ross recovery theorem, which also exploits the risk-neutral information of the state variable. We assume that the state variable is a one-dimensional time-homogeneous Markov diffusion and the utility function is an increasing, strictly concave function satisfying the Inada conditions. Under these assumptions, the primary contributions of this study are as follows: First, we provide a necessary and sufficient condition for the existence of a utility function in terms of the Feller boundary classification of the state variable. We show that a utility function exists if and only if the left boundary of the state variable is a natural boundary. Second, if we additionally assume that the state variable is non-attracted to the left boundary, then we show that the utility function can be described as a one-parameter model. In the specific case in which the short interest rate is a constant, the utility functions are fully described. Third, our method is used for Ross recovery. Last, we provide examples with explicit solutions.

Title : A BSDE approach to the large-time optimal expected utility in incomplete markets

Abstract : In this study, we are interested in the long-term expected utility of optimal portfolios for an investor. Under an incomplete market given by a factor model on the Euclidean state space, we consider the utility maximization problem with long-time horizon. Using the duality method and the control theory, this problem can be solved by analyzing the long-time behavior of solutions to semi-linear parabolic PDEs with quadratic term in gradients. Under certain conditions on the market price of risk, we conclude that the parabolic PDE induces an eigenpair which characterizes the long-term expected utility of optimal portfolios. To achieve this, we apply BSDE techniques and PDE theories to an approximated PDE and show that solutions of the approximated PDE converge to the eigenpair. This is a joint work in progress with Hyungbin Park and Stephan Sturm.

Title: Minimum curvature flow and martingale exit times

Abstract: We study the following question: What is the largest deterministic amount of time T∗ that a suitably normalized martingale X can be kept inside a convex body K in Rd? We show, in a viscosity framework, that T∗ equals the time it takes for the relative boundary of K to reach X(0) as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio–Soner codimension-(d − 1) mean curvature flow of the 1-skeleton of K. We present very preliminary sampling-based numerical approximations to the solution of the corresponding PDE. The numerical part is work in progress. This work is based on a collaboration with Camilo Garcia Trillos, Martin Larsson, and Yufei Zhang.

Title: Total variation bounds for Milstein scheme and Euler-Maruyama scheme: application to mathematical finance

Abstract: In the talk, we give new results on Milstein and Euler-Maruyama schemes for stochastic differential equations. In particular, we show that a version of Milstein scheme given by second order polynomials of Brownian motion without using iterated integrals holds as a weak approximation in total variation sense under non-commutative vector fields, and show that the accuracy of the extended Milstein scheme is better than that of the Euler-Maruyama scheme in a small noise asymptotic sense. The extended Milstein scheme may be interpreted as an extension of the schemes proposed in Cruzeiro, Malliavin and Thalmaier (2004) and Davie (2014). In computational aspect, the extended Milstein scheme can be applied to the estimation of probability distribution functions by a simple simulation without L\'evy area computation. Numerical examples for finance models demonstrate the validity of the theoretical results.

Title: Large-time behavior of optimal portfolios and their sensitivities in non-affine models

Abstract: We investigate a large-time behavior of the optimal portfolio and its sensitivity in non-affine models. An investor maximizes the expected utility with constant relative risk aversion under an incomplete market consisting of a safe asset, several risky assets, and a single non-affine state process. Two specific non-affine state models are covered: 3/2 process and inverse Bessel process. We first show that the optimal portfolio converges to a time-homogeneous portfolio that depends only on the current state variable. The time-homogeneous portfolio and the convergence rate are computed explicitly for the two non-affine models. To show this, the Hansen-Scheinkman factorization is used as a main tool. We also conduct a sensitivity analysis of the optimal portfolio and investigate its large-time asymptotic behavior.

Title: Design of Market-Clearing Technology

Abstract: Advances in technology have increased interest in innovations that clear multiple assets-traditionally, assets have cleared independently. This paper examines the problem of designing multi-asset market clearing with a rich family of cross-asset demand conditioning rules. Suitably designed market-clearing technology improves efficiency relative to independent clearing of all assets under general conditions. Depending on market characteristics, efficient design requires different disclosure and possibly withholding of past data from past- and current rounds. Our results qualify in which markets' investment in multi-asset market-clearing technology is worthwhile. We identify a subclass of designs that can equivalently be implemented dynamically without requiring an investment in multi-asset technology. This work is based on a collaboration with M. Rostek and C. Lyu.